Characterization of Contaminant Concentrations in Enclosed Spaces Nurtan A. Esmen Department of Industrial Environmental Health Sciences, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, Pa. 15261
Generalized equations for contaminant concentration buildup and decay are developed for an enclosed, ventilated space. The generalized equations take local mixing factors and source characteristics into account, and the illustrative cases show that the usual method of estimation can be totally erroneous. The estimation of transient and equilibrium concentrations of contaminants in enclosed, ventilated spaces is important for both personal comfort and safety of the occupants. Based on the solution of the mass balance equation, if the air in the room under consideration is well mixed, the decrease of pollutant after a single release will be an exponential decay. The nonideality of mixing is usually accounted for by the utilization of a mixing factor, h . The values of k are normally estimated to be from '/3 to I/~o for small spaces (1-31, and it is possibly lower for very large spaces. The experimental verification of the concept of the mixing factor in the exponential decay has been excellent. The traditional physical meaning attached to the mixing factor is the ratio of the characteristic time for one theoretical air change to the characteristic time for one actual air change as calculated from the experimental results. This concept can lead to theoretical difficulties and indeed to erroneous results if continuous or cyclic sources are present in the room under consideration. Instead of the above-mentioned, a physical description of the mixing factor may be proposed without the alteration of its numerical value. In this paper such a definition will be proposed. In addition, if the room contains multiple sources that operate periodically, then the usual estimation procedure is to equate the generation rate to a single, continuous source by averaging the emission from all of the sources. In this procedure, two important facets of the exposure to the contaminant are overlooked. The procedure does not indicate what the peak concentrations might be; further, the equilibrium concentration calculated is not representative of the mean concentration as will be shown in this paper. Theoretical Considerations Consider a room with volume V ,a ventilation air flow rate of Q, and an initial concentration of a contaminant Ci,. The concentration of the contaminant is described by the mass balance, assuming ideally well-mixed conditions and no reaction:
the solution of Equation 1 is:
The factor in the exponent that describes the theoretical characteristic time for one air change is VlQ, and traditionally the mixing factor, k , is described as: h = -VlQ
(3)
exp
0013-936X/78/0912-0337$01.00/0
0 1978 American
Chemical Society
With the use of this factor, Equation 2 becomes: (4)
Note that Equation 4 will satisfy the mass balance only for h = 1 or under ideally mixed conditions. Now, let h be defined as the portion of the ventilation air flow, Q , that is completely mixed with the room air. In other words, we imagine that the h fraction of the air flowing in will be totally replaced by the room air, and the remainder is totally unmixed. Under this condition the mass balance is:
(5) and the solution of Equation 5 is identical to Equation 4:
kQt c = ci, exp (- 7) The numerical value of h given in the definition above is identical to the experimentally determined value, and the final equation satisfies the mass balance for all values of h . Now we can generalize this concept to include any number of sources in the room. In the development of the general equation that describes the concentration of a specific contaminant in a room of volume V and for a ventilation volumetric flow rate of Q , the following assumptions will be made. I t will be assumed that there are a number of sources of the contaminant in question, and that each source will have a time dependent generation rate, G,,, for the ith time interval for t h e j t h source. In reality the generation rate for a given source in a given time interval is not constant; however, the function that describes the generation rate is usually not known and is difficult to obtain. Therefore, the generation rate, GI,, will be assumed to be constant over a given interval, i, and for a given source, j . The air supply system that defines the ventilation of the room in question may have a portion of the air supplied from ambient air and a portion recirculated. The recirculated portion may pass through a collector. Therefore, it will be assumed that the ventilation flow rate is composed of a portion that is recirculated, QR, and a portion that is supplied from the outdoor air, Q,. I t will be further assumed that a collection device of efficiency E is present in the recirculation loop, a concentration of C, of the contaminant is present in the makeup air. The room has an initial concentration of C,, and a known mixing factor k . It is possible to develop the controlling equations for a single periodic source in the room and then to generalize it to any number of sources periodic, or otherwise, in the final form. Consider a source with generation rates C, for time intervals i and let the time interval duration J , be such that during a , fraction there is generation and 1 - a Lfraction inactivity. The mass balance for the room is: dC = G ( t ) - kQoC - kQREC kQ,C, (7) dt Equation 7 does not include a reactive term, i.e., it is tacitly assumed that the contaminant is conservative. The solution
V-
+
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337
of Equation 7 for completion of n cycles may be shown to be:
Similarly,
Equations 16 and 17 can be further reduced for conditions when @;e > 200 a, and @,e> 20. This condition is sufficiently near the equilibrium value where P;8 m so that the deviation from the equilibrium is less than 5%:
-
where
and (12) In Equation 8 the local mixing is adjusted by the definition of a mixing factor ki for each cycle and a mixing factor for the entire space, h,, where the source has no contribution. If a number of sources are present in the room then each source, j , will have n,, cycles over the time period, t ; hence:
In Equation 13 the double subscripted parameters contain k,,, all, G,,, and J1,and the definitions of these parameters are obtained by substituting the double subscripted variables for the single subscripted variables in Equations 9-12. The numerical evaluation of Equation 13 involves the knowledge of the parameters of each individual cycle for each individual source. However, in practice, this is neither tractable nor necessary. What is of importance is an estimate of the average concentration and an estimate of the peak concentration. The average can be obtained for a time 8 > 0 by:
Now Equation 14 may be simplified to a tractable form (albeit approximate) by the assumption that the properties of each cycle may be satisfactorily expressed with an average value for each source. Since n;J; = 8 then, it may be shown that for 8 > J :
(15)
Discussion of Results Note that in the expressions developed above, one has a complete generality: a, = 0 implies that there is no emission from the source; a; = 1 implies that the source is continuous. Furthermore, the P values are locally calculated for each source. The local calculation of the parameters for each source is especially important for large rooms. This can be illustrated by a simple, realistic example: a person who does not wish to be exposed to intense tobacco smoke from a smoker can move away from the smoker. This is equivalent to the nonsmoker’s choosing a place with a higher k value with respect to the nearest smoker. The equations developed in this paper can now be compared to the usual expression for the contaminant concentration. In the equation generally used, all of the sources are assigned to one overall generation factor, and the equation does not involve the k factor in any place besides the decay exponent ( 4 ) :
This equation implies that the equilibrium concentration a t any place in the room will be the same regardless of the mixing factor. This condition is erroneous because if it is true, then the mixing factor must be uniform, i.e., k = 1.As a simple illustration, the case of smoker and nonsmoker can be used again. Since both the smoker and nonsmoker are in the same room by Equation 20, both the smoker and the nonsmoker are exposed to the same concentration. This is obviously incorrect; for the smoker, k , is much lower than for the nonsmoker. Another extreme example can be given by a warm room where cold air intake and exhaust are located very near the floor; the mixing under this circumstance will be very poor, and in fact the equilibrium concentration of a contaminant emitting from a high source will be quite high. According to Equation 20, relocation of the intake and exhaust and utilization of proper diffusers to mix the air will result in the same equilibrium concentration. This again is obviously incorrect. Furthermore, Equation 20 will result in lower values than the equations developed here, hence leading to optimistic estimates that can prove to be hazardous to the health of the occupants. Nomenclature
C = contaminant concentration C, = average contaminant concentration Ci, = initial contaminant concentration 338
Environmental Science & Technology
C, = contaminant concentration in the incoming air
E = efficiency of the collector G,, = generation rate of j t h source a t i t h interval J , = duration of ith interval I’ = 1 - exp(-$tuJ) Q = volumetric flow rate Q , = volumetric flow rate of makeup air QH = volumetric flow rate of recirculated air V = roomvolume K = mixing factor A4 = total number of sources n, = total number of cycles for j t h source over fl time t = time
$ = decay coefficient 0 = averaging time Subscripts
i = time interval index j = source index Literature Cited (1) Brief, R. S., Air Eng., 2,39 (1960). ( 2 ) Constance, J. D., Power, 114, 56-7 (1970). ( 3 ) Drivas. P. J . , Simmonds, P. G., Shair, F. H., Enciron. Sei. Technol., 6,609-14 (1972). (4)Turk, A,, A S H R A E J., 5,55-8 (1963).
Greek Letters = fraction of time for generation
Rcceiued for review February 13, 1977. Accepted Septpmber 29, 1977.
CORRESPONDENCE
Equilibrium constants are determined as follows. The standard Gibbs energy change accompanying a chemical process, AGO, is obtained by the relationship:
CY
SIR: Studies of problems in the area of water quality cont rol, the application of geothermal energy, the desalinization of water, sewage treatment, and bioengineering all must treat aqueous solutions containing ionic species. An enormous need for reliable quantitative data on ionic equilibria has become very apparent in recent years, particularly with the development of large scale models that attempt to simulate complex aqueous ecosystems ( I 1. The purposes of this letter are to point out that many equilibrium constants involving ions in solution can be extracted from existing tables of evaluated thermochemical data, and to provide three tables of data applicable to studies of the water environment. The tables give thermodynamic equilibrium constants for the association of alkaline earth cations with various ligands and the thermodynamic properties of a number of other species. From the latter the user can construct equilibrium constants pertinent to specific studies. The method used to obtain the equilibrium constants is summarized as an aid to the user. Two comprehensive series of thermochemical tables contain the data of interest. These are “Selected Values of Chemical Thermodynamic Properties” (2-7) issued by the U.S. National Bureau of Standards and “Termicheskie Konstanty Veschestv” (8)issued by the Institute of High Temperatures of the Academy of Sciences of the USSR. Both tabulate thermodynamic properties of individual substances from which those for particular processes can be derived. The properties of interest that are tabulated here are the Gibbs (free) energy of formation of a substance from the elements, A G f O , the enthalpy of formation, Sfo, the absolute entropy, So,and the heat capacity, C,, all a t 298.15 K. Typical defining reactions for formation processes of a salt and of an ion are, respectively: 2Na(c) Ca(c)
+ S(c) + 202(g) = NasSO4(c)
+ 2H+(aq) = Ca2+(aq)+ H,(g)
where (c) is condensed phase, ( a s ) is aqueous, and (g) is gas. The advantage of using either of these tables as a starting point for the construction ofequilibrium constants lies in the fact that the data therein are evaluated and self-consistent. In other words, experts have decided which experimental data are likely to be most reliable and have arranged them on a common scale. This relieves the user of the necessity of deciding which of several often discordant results should be used.
AGO = ~ v v , ( A C , f o-) , C v l ( A G f o ) ,
(1)
products reactants where the superscript O refers to standard state conditions, the f identifies the formation process, and the vL’s are the stoichiometric coefficients of the reaction. The standard Gibbs energy change is directly related to the equilibrium constant, K , via A G O
=
-RT In K
(2)
where R is the gas constant, T is the absolute temperature in Kelvins, and In denotes the logarithm to the base e . As an illustration, the equilibrium constant is calculated for the process forming an ion pair, MgOH+, in aqueous solution, a t 298.15 K: Mg2+
+ OH-
s MgOH+
(3)
The data needed are listed in Tables 1-111 and can be obtained from refs. 2-7. Table I presents the data for selected species of alkaline earth salts, and Tables I1 and I11 are provided as a ready reference for Gibbs energies of formation for selected cations (Table 11) and anions (Table 111). Table I lists various common dissolved species, thought to exist in the aqueous environment, the formula weight, the Sv, (AGf), for the product (ion pair) and the reactants (individual ions), and the AGO for the reaction (see Equation 1).All thermodynamic functions are expressed both in kcal-mol-’ and in kJ-mol-1. Also listed are the association equilibrium constants and their logarithms to the base 10. Tables I1 and I11 simply provide the AGfO of the ions which are used to derive AGO for a reaction by means of Equation 1. Tables I1 and I11 provide a selection of ions found scattered throughout the Tech. Note 270-series (refs. 2-7) and may be used in conjunction with this 270-series, thus eliminating the need for a comprehensive search for individual, simple ionic species. For Reaction 3 using Equation 1: AGO =
A G f O
(MgOH+)-
[ A G f O
+
(Mg2+) AGf” (OH-)] (4)
Thus, A G O
- 37.594) and AGO = -3.506 kcal . mol-]
= -149.8 - (-108.7
(5)
Then, from Equations 2 and 3 and the definition of the thermodynamic equilibrium constant, K
This article not subject to US. Copyright. Published 1978 American Chemical Society
Volume 12, Number 3, March 1978
339
